Table Of ContentInstability in Models
Connected with Fluid
Flows II
INTERNATIONAL MATHEMATICAL SERIES
SeriesEditor: TamaraRozhkovskaya
SobolevInstituteofMathematicsoftheSiberianBranch
oftheRussianAcademyofSciences,Novosibirsk,Russia
1. NonlinearProblemsinMathematicalPhysicsandRelatedTopicsI.InHonor
ofProfessorO.A.Ladyzhenskaya•M.Sh.Birman,S.Hildebrandt,V.A.Solon-
nikov,N.N.UraltsevaEds.•2002
2. Nonlinear Problems in Mathematical Physics and Related Topics II. In
Honor of Professor O.A. Ladyzhenskaya • M.Sh. Birman, S. Hildebrandt,
V.A.Solonnikov,N.N.UraltsevaEds.•2003
3. DifferentFacesofGeometry•S.Donaldson,Ya.Eliashberg,M.GromovEds.
•2004
4. MathematicalProblemsfromAppliedLogicI.LogicsfortheXXIstCentury
•D.Gabbay,S.Goncharov,M.ZakharyaschevEds.•2006
5. MathematicalProblems from Applied Logic II. Logics for the XXIst Cen-
tury•D.Gabbay,S.Goncharov,M.ZakharyaschevEds.•2007
6. InstabilityinModelsConnectedwithFluidFlowsI•C.Bardos,A.Fursikov
Eds.•2008
7. InstabilityinModelsConnectedwithFluidFlowsII•C.Bardos,A.Fursikov
Eds.•2008
Instability in Models
Connected with Fluid
Flows II
Editedby
Claude Bardos
UniversitéDenisDiderot
LaboratoireJ.-L.Lions,Université6
Paris,France
Andrei Fursikov
MoscowStateUniversity
InstituteofNumericalMathematicsRAS
Moscow,Russia
Editors
ClaudeBardos AndreiFursikov
UniversitéDenisDiderot MoscowStateUniversity
LaboratoireJ.-L.Lions,Université6 InstituteofNumericalMathematicsRAS
Paris,France Moscow,Russia
This series was founded by Kluwer / Plenum Publishers (now Springer) and the Russian
publisherTamaraRozhkovskaya(Novosibirsk,Russia,[email protected])in2002.Each
volumepresentscontributionsfromvolumeeditorsandauthorsinvitedbytheSeriesEditor.
TheEnglishCamera-ReadyManuscriptwaspreparedbyTamaraRozhkovskaya.
LibraryofCongressControlNumber:2007938320
ISBN:978-0-387-75218-1 e-ISBN:978-0-387-75219-8
ISSN:1571-5485
(cid:2)c 2008SpringerScience+BusinessMedia,LLC
Allrightsreserved. Thisworkmaynot betranslatedorcopied inwholeor inpart without
thewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,LLC,233Spring
Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews
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Instability in Models Connected with Fluid Flows I, II
Two volumesof the International Mathematical Series presentvarioustop-
icsoncontroltheory,freeboundaryproblems,theNavier–Stokesequations,
attractors, first order linear and nonlinear equations, partial differential
equations of fluid mechanics, etc. with the focus on the key question in the
study of mathematical models simulating physical processes:
Is a model stable (or unstable) in a certain sense?
An answer provides us with understanding the following issue, extremely
important for applications:
Does the model adequately describe the physical process?
Recent advantages in this area, new results, and current approaches to the
notion of stability are presented by world-recognizedexperts.
Main Topics
• Navier-Stokes equations. Existence and smoothness results
— Localandglobalexistenceresultsforthe3–dimensionalNavier-Stokes
system without external forcing when the initial conditions are the
Fourier transforms of finite-linear combinations of δ–functions.
Efim Dinaburg and Yakov Sinai, Vol. I
— The analyticity of periodic solutions of the 2D Boussinesq system.
Maxim Arnold, Vol. I
— Navier–Stokes equations in cylindrical domains. Leray approxima-
tions, Leray–Navier–Stokes equations, the Helmholtz projector and
stationary Stokes problem, the classical Navier–Stokes problem.
Sergey Zelik, Vol. II
• First order linear and nonlinear equations
— Nonlinear dynamics of a system of particle-like wavepackets, reduc-
tionofwavepacketinteractionsystemstoaveragedones,superposition
principle and decoupling of wavepacketinteraction systems.
Anatoli Babin and Alexander Figotin, Vol. I
— Transport equations with discontinuous coefficients, Keyfitz-Kranzer
typehyperbolicsystems,generalizedsolutionsoftheCauchyproblem,
existence, uniqueness, and renormalizationproperty.
Evgenii Panov, Vol. II
— Navier–Stokes approximations, moment approximations of the Boltz-
mann–Peierlskineticequation,Chapman–Enskogprojectionsofdiffu-
sion and boundary-layer type, the mixed problem.
Evgenii Radkevich, Vol. II
viii Main Topics
• Finite time instabilities of Euler equations
— Large amplitude monophase nonlinear geometric optics, the case of
incompressible Euler equations, large amplitude waves.
Christophe Cheverry, Vol. I
— Bursting dynamics of the 3D Euler equations in cylindrical domains,
vorticity waves, strictly resonant Euler systems.
Francois Golse, Alex Mahalov, and Basil Nicolaenko, Vol. I
• Large time asymptotics of solutions
— Attractors for the Navier–Stokes system, autonomous and nonauto-
nomous equations, the Kolmogorovε-entropyof globalattractors,2D
Navier–Stokes equations, the Ginzburg–Landau equation.
Vladimir Chepyzhov and Mark Vishik, Vol. I
• Statistical approach
— Exponential mixing for randomly forced partial differential equations
(method of coupling), Markov random dynamical system, dissipative
random dynamical systems, the complex Ginzburg–Landau equation.
Armen Shirikyan, Vol. II
• Water waves and free boundary problems
— Asymptotics for 3D water–waves, large time existence theorems, the
Kadomtsev–Petviashviliiapproximation.
David Lannes, Vol. II
— Stabilityofarotatingcapillaryviscousincompressibleliquidbounded
by a free surface.
Vsevolod Solonnikov, Vol. II
— Symmetric compressible barotropic Navier–Stokes-Poisson flows in a
vacuum, the existence of global weak solutions.
Alexander Zlotnik, Vol. II
• Optimal control
— Increasedstability in the Cauchy problem for some elliptic equations,
energy type estimates in low frequency zone, Carleman estimates.
Victor Isakov, Vol. I
— Controllability and accessibility of equations of dynamics of incom-
pressible fluids controlled by low-dimensional (degenerate) forcing,
controllabilityofNavier–Stokes/Eulerequationsonatwo-dimensional
sphere and on a generic Riemannian surface.
Andrey Agrachev and Andrey Sarychev, Vol. I
Editors
Claude Bardos
Professeur Emeritus
UniversiteDenis Diderot
Laboratoire Jacques-Louis Lions
Universit´eParis 6
75252 Paris Cedex 05
France
[email protected]
http://www.ann.jussieu.fr
Andrei Fursikov
Professor
Moscow StateUniversity
Moscow 119992
Instituteof Numerical
Mathematics of theRussian
Academy of Sciences
8, ul. Gubkina
Moscow 119991
Russia
[email protected]
http://mech.math.msu.su/˜fursikov
PhotosofC.Bardos(Photographer: Andrea Baczynski,Austria)andA.Fursikov
(family’s archive) are reproduced underpermission.
Authors
Andrey Agrachev
International School for Advanced Studies
4, via Beirut
34014 Trieste
ITALY
[email protected]
Steklov Mathematical Institute
Russian Academy of Scineces
8, ul. Gubkina
Moscow 119991
RUSSIA
[email protected];
Maxim Arnold
International Institute of Earthquake
Prediction Theory and Mathematical Geophysics
Russian Academy of Sciences
k.2, 79, Warshavskoyesh.
Moscow 117556
RUSSIA
[email protected]
xii Authors
Anatoli Babin
University of California,
Irvine
CA 92697-3875
USA
[email protected]
Vladimir Chepyzhov
Institute for Information Transmission Problems
Russian Academy of Sciences
19, Bol’shoi Karetnyi pereulok
Moscow 127994
RUSSIA
[email protected]
Christophe Chevery
Universit´e de Rennes I
Campus de Beaulieu
35042 Rennes
FRANCE
[email protected]
Efim Dinaburg
International Institute of Earthquake
Prediction Theory and Mathematical Geophysics
Russian Academy of Sciences
k. 2, 79, Warshavskoyesh.
Moscow 117556
RUSSIA
Francois Golse
Ecole Polytechnique
Centre de math´ematiques Laurent Schwartz
91128 PalaiseauCedex
FRANCE
[email protected]
